The finite element method in heat transfer and fluid dynamics

Title:

Personal Author:

Reddy, J.N.

Publication Information:

United kingdom : Taylor & Francis, 2000

ISBN:

9780849323553

Subject Term:

Fluid dynamics -- Mathematical models

Heat -- Transmission -- Mathematical models

Finite element method

Added Author:

Gartling, David K.

Available:*

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The numerical simulation of fluid mechanics and heat transfer problems is now a standard part of engineering practice. The widespread availability of capable computing hardware has led to an increased demand for computer simulations of products and processes during their engineering design and manufacturing phases. The range of fluid mechanics and heat transfer applications of finite element analysis has become quite remarkable, with complex, realistic simulations being carried out on a routine basis.

The award-winning first edition of The Finite Element Method in Heat Transfer and Fluid Dynamics brought this powerful methodology to those interested in applying it to the significant class of problems dealing with heat conduction, incompressible viscous flows, and convection heat transfer. The Second Edition of this bestselling text continues to provide the academic community and industry with up-to-date, authoritative information on the use of the finite element method in the study of fluid mechanics and heat transfer. Extensively revised and thoroughly updated, new and expanded material includes discussions on difficult boundary conditions, contact and bulk nodes, change of phase, weighted-integral statements and weak forms, chemically reactive systems, stabilized methods, free surface problems, and much more.

The Finite Element Method in Heat Transfer and Fluid Dynamics offers students a pragmatic treatment that views numerical computation as a means to an end and does not dwell on theory or proof. Mastering its contents brings a firm understanding of the basic methodology, competence in using existing simulation software, and the ability to develop some simpler, special purpose computer codes.

Author Notes

J. N. Reddy earned a Ph.D. in Engineering Mechanics and worked as a Postdoctoral Fellow at the University of Texas at Austin, Research Scientist for Lockheed Missiles and Space Company during 1974-75, and taught at the University of Oklahoma from 1975 to 1980 and Virginia Polytechnic Institute and State University from 1980 to 1992. Currently, he is a Distinguished Professor and the inaugural holder of the Oscar S. Wyatt Endowed Chair at Texas AandM University
David K. Gartling is a Senior Scientist in the Engineering Sciences Center at Sandia National Laboratories, Albuquerque, New Mexico. He earned his B.S. and M.S. in Aerospace Engineering at the University of Texas at Austin and completed the diploma course at the von Karman Institute for Fluid Dynamics in Brussels, Belgium. After completion of his Ph.D. in Aerospace Engineering at the University of Texas at Austin, he joined the technical staff at Sandia National Laboratories. Dr. Gartling was a Visiting Associate Professor in the Mechanical Engineering Department at the University of Sydney, Australia under a Fulbright Fellowship, and later he was a Supervisor in the Fluid and Thermal Sciences Department at Sandia National Laboratories

Preface to the Second Edition	p. v
Preface to the First Edition	p. vii
1. Equations of Heat Transfer and Fluid Mechanics	p. 1
1.1 Introduction	p. 1
1.1.1 Heat Transfer	p. 1
1.1.2 Fluid Mechanics	p. 2
1.2 Present Study	p. 3
1.3 Governing Equations of a Continuum	p. 3
1.3.1 Introduction	p. 3
1.3.2 Conservation of Mass; the Continuity Equation	p. 4
1.3.3 Conservation of Momenta	p. 4
1.3.4 Conservation of Energy	p. 5
1.3.5 Constitutive Equations	p. 5
1.4 Governing Equations in Terms of Primitive Variables	p. 7
1.4.1 Vector Form	p. 7
1.4.2 Cartesian Component Form	p. 7
1.4.3 Cylindrical Component Form	p. 8
1.4.4 Closure	p. 9
1.5 Porous Flow Equations	p. 10
1.6 Auxiliary Transport Equations	p. 11
1.7 Chemically Reacting Systems	p. 12
1.8 Boundary Conditions	p. 15
1.8.1 Viscous Flow Boundary Conditions	p. 15
1.8.2 Porous Flow Boundary Conditions	p. 18
1.8.3 Thermal and Transport Boundary Conditions	p. 19
1.8.4 Initial Conditions	p. 20
1.9 Change of Phase	p. 21
1.10 Enclosure Radiation	p. 23
1.11 Summary of Equations	p. 25
Problems	p. 26
References for Additional Reading	p. 28
2. The Finite Element Method: An Overview	p. 31
2.1 Introduction	p. 31
2.2 Model Differential Equation	p. 32
2.3 Finite Element Approximation	p. 33
2.4 Weighted-Integral Statements and Weak Forms	p. 35
2.5 Finite Element Model	p. 38
2.6 Interpolation Functions	p. 39
2.7 Assembly of Elements	p. 43
2.8 Time-Dependent Problems	p. 45
2.8.1 Introduction	p. 45
2.8.2 Semidiscretization	p. 46
2.8.3 Temporal Approximation	p. 47
2.9 Axisymmetric Problems	p. 48
2.10 Convective Boundary Conditions	p. 51
2.11 Library of Finite Elements	p. 52
2.11.1 Introduction	p. 52
2.11.2 Triangular Elements	p. 52
2.11.3 Rectangular Elements	p. 54
2.12 Numerical Integration	p. 55
2.12.1 Preliminary Comments	p. 55
2.12.2 Coordinate Transformations	p. 57
2.12.3 Integration Over a Master Rectangular Element	p. 60
2.12.4 Integration Over a Master Triangular Element	p. 61
2.13 Modeling Considerations	p. 62
2.13.1 Mesh Generation	p. 62
2.13.2 Representation of Boundary Flux	p. 64
2.13.3 Imposition of Boundary Conditions	p. 64
2.14 Illustrative Examples	p. 65
2.14.1 Example 1	p. 66
2.14.2 Example 2	p. 72
2.14.3 Example 3	p. 73
Problems	p. 74
References for Additional Reading	p. 77
3. 3-D Conduction Heat Transfer	p. 79
3.1 Introduction	p. 79
3.2 Semidiscrete Finite Element Model	p. 80
3.3 Interpolation Functions	p. 83
3.3.1 Preliminary Comments	p. 83
3.3.2 Hexahedral (Brick) Elements	p. 83
3.3.3 Prism Elements	p. 85
3.3.4 Tetrahedral Elements	p. 86
3.4 Numerical Integration	p. 87
3.5 Computation of Surface Flux	p. 88
3.6 Semidiscrete Finite Element Model	p. 91
3.7 Solution of Nonlinear Equations	p. 92
3.7.1 Preliminary Comments	p. 92
3.7.2 Steady-State Problems	p. 92
3.7.3 Transient Problems	p. 94
3.8 Radiation Solution Algorithms	p. 104
3.9 Variable Properties	p. 108
3.9.1 Temperature-Dependent Properties	p. 108
3.9.2 Phase Change Properties	p. 109
3.9.3 Anisotropic Properties	p. 111
3.10 Post-Processing Operations	p. 112
3.10.1 Heat Flux	p. 112
3.10.2 Heat Flow Function	p. 114
3.11 Advanced Topics in Conduction	p. 115
3.11.1 Introduction	p. 115
3.11.2 Specialty Elements	p. 116
3.11.3 Computational Boundary Conditions	p. 119
3.11.4 Bulk Nodes	p. 125
3.11.5 Reactive Materials	p. 127
3.11.6 Material Motion	p. 129
3.12 Example Problems	p. 130
3.12.1 Introduction	p. 130
3.12.2 Temperature-Dependent Conductivity	p. 131
3.12.3 Anisotropic Conductivity	p. 131
3.12.4 One-Dimensional Stefan Problem	p. 133
3.12.5 Drag Bit Analysis	p. 135
3.12.6 Brazing and Welding Analyses	p. 136
3.12.7 Investment Casting	p. 141
Problems	p. 143
References for Additional Reading	p. 144
4. Viscous Incompressible Flows	p. 149
4.1 Introduction	p. 149
4.1.1 Background	p. 149
4.1.2 Governing Equations	p. 149
4.2 Mixed Finite Element Model	p. 152
4.2.1 Weak Form	p. 152
4.2.2 Finite Element Model	p. 153
4.3 Penalty Finite Element Models	p. 156
4.3.1 Introduction	p. 156
4.3.2 Penalty Function Method	p. 157
4.3.3 Reduced Integration Penalty Model	p. 159
4.3.4 Consistent Penalty Model	p. 160
4.4 Finite Element Models of Porous Flow	p. 161
4.5 Computational Considerations	p. 163
4.5.1 Properties of the Matrix Equations	p. 163
4.5.2 Choice of Interpolation Functions	p. 164
4.5.3 Evaluation of Element Matrices in Penalty Models	p. 169
4.5.4 Pressure Calculation	p. 170
4.5.5 Traction Boundary Conditions	p. 172
4.6 Solution of Nonlinear Equations	p. 175
4.6.1 General Discussion	p. 175
4.6.2 Fully Coupled Solution Methods	p. 178
4.6.3 Pressure Correction/Projection Methods	p. 183
4.7 Time-Approximation Schemes	p. 186
4.7.1 Preliminary Comments	p. 186
4.7.2 Forward/Backward Euler Schemes	p. 186
4.7.3 Adams-Bashforth/Trapezoid Rule	p. 187
4.7.4 Implicit Integration and Time Step Control	p. 188
4.7.5 Explicit Integration	p. 189
4.8 Stabilized Methods	p. 190
4.8.1 Preliminary Comments	p. 190
4.8.2 Galerkin/Least-Squares Formulation	p. 191
4.9 Post-Processing	p. 194
4.9.1 Stress Computation	p. 194
4.9.2 Stream Function Computation	p. 196
4.9.3 Particle Tracking	p. 198
4.10 Advanced Topics - Free Surface Flows	p. 198
4.10.1 Preliminary Comments	p. 198
4.10.2 Time-Independent Free Surfaces	p. 199
4.10.3 Time-Dependent Free Surfaces	p. 204
4.11 Advanced Topics - Turbulence	p. 211
4.11.1 Preliminary Comments	p. 211
4.11.2 Governing Equations	p. 212
4.11.3 General Turbulence Models	p. 213
4.11.4 One-Point Closure Turbulence Models	p. 215
4.11.5 Finite Element Modeling of Turbulence	p. 219
4.12 Numerical Examples	p. 221
4.12.1 Preliminary Comments	p. 221
4.12.2 Fluid Squeezed between Parallel Plates	p. 222
4.12.3 Flow of a Viscous Lubricant in a Slider Bearing	p. 225
4.12.4 Wall-Driven 2-D Cavity Flow	p. 226
4.12.5 Wall-Driven 3-D Cavity Flow	p. 229
4.12.6 Evaluation of the EBE Iterative Solvers	p. 229
4.12.7 Backward Facing Step	p. 233
4.12.8 Flow Past a Submarine	p. 235
4.12.9 Crystal Growth from the Melt	p. 237
4.12.10 Mold Filling	p. 238
Problems	p. 242
References for Additional Reading	p. 243
5. Convective Heat Transfer	p. 255
5.1 Introduction	p. 255
5.1.1 Background	p. 255
5.1.2 Governing Equations	p. 255
5.2 Mixed Finite Element Model	p. 257
5.3 Penalty Finite Element Model	p. 261
5.3.1 Preliminary Comments	p. 261
5.3.2 Reduced Integration Penalty Model	p. 262
5.3.3 Consistent Penalty Model	p. 263
5.4 Finite Element Models of Porous Flow	p. 263
5.5 Solution Methods	p. 265
5.5.1 General Discussion	p. 265
5.5.2 Newton's Method	p. 266
5.5.3 Segregated Equation Methods	p. 267
5.6 Convection with Change of Phase	p. 269
5.7 Convection with Enclosure Radiation	p. 271
5.8 Post-Computation of Heat Flux	p. 271
5.9 Advanced Topics - Turbulent Heat Transfer	p. 273
5.10 Advanced Topics - Chemically Reacting Systems	p. 274
5.10.1 Preliminary Comments	p. 274
5.10.2 Finite Element Modeling of Chemical Reactions	p. 274
5.11 Numerical Examples	p. 275
5.11.1 Preliminary Comments	p. 275
5.11.2 Concentric Tube Flow	p. 275
5.11.3 Tube Flow with Change of Phase	p. 276
5.11.4 Heated Cavity	p. 278
5.11.5 Solar Receiver	p. 279
5.11.6 Tube Bundle	p. 282
5.11.7 Volumetrically Heated Fluid	p. 284
5.11.8 Porous/Fluid Layer	p. 287
5.11.9 Curing of An Epoxy	p. 289
References for Additional Reading	p. 292
6. Non-Newtonian Fluids	p. 295
6.1 Introduction	p. 295
6.2 Governing Equations of Inelastic Fluids	p. 296
6.2.1 Conservation Equations	p. 296
6.2.2 Boundary Conditions	p. 297
6.2.3 Constitutive Equations	p. 298
6.3 Finite Element Models of Inelastic Fluids	p. 301
6.3.1 Introduction	p. 301
6.3.2 Mixed Model	p. 302
6.3.3 Penalty Model	p. 304
6.3.4 Matrix Evaluations	p. 305
6.4 Solution Methods for Inelastic Fluids	p. 307
6.5 Governing Equations of Viscoelastic Fluids	p. 311
6.5.1 Conservation Equations	p. 311
6.5.2 Constitutive Equations	p. 312
6.5.3 Boundary Conditions	p. 318
6.6 Finite Element Model of Differential Form	p. 318
6.6.1 Preliminary Comments	p. 318
6.6.2 Summary of Governing Equations	p. 318
6.6.3 Finite Element Model	p. 320
6.6.4 Solution Methods	p. 324
6.7 Additional Models of Differential Form	p. 325
6.7.1 Explicitly Elliptic Momentum Equation Method	p. 326
6.7.2 Elastic Viscous Stress Splitting Method	p. 327
6.8 Finite Element Model of Integral Form	p. 329
6.9 Unresolved Problems	p. 331
6.9.1 General Comments	p. 331
6.9.2 Choice of Constitutive Equation	p. 332
6.9.3 Solution Uniqueness and Existence	p. 332
6.9.4 Numerical Algorithm Problems	p. 333
6.9.5 Equation Change of Type	p. 334
6.9.6 Closure	p. 335
6.10 Numerical Examples	p. 335
6.10.1 Preliminary Comments	p. 335
6.10.2 Buoyancy Driven Flow in a Cavity	p. 336
6.10.3 Driven Cavity Flow	p. 336
6.10.4 Squeeze Film Flow	p. 339
6.10.5 Time-Dependent Poiseuille Flow	p. 341
6.10.6 Four-to-One Contraction Problem	p. 343
Problems	p. 346
References for Additional Reading	p. 346
7. Coupled Problems	p. 353
7.1 Introduction	p. 353
7.2 Coupled Boundary Value Problems	p. 353
7.3 Fluid Mechanics and Heat Transfer	p. 354
7.3.1 Introduction	p. 354
7.3.2 Continuum Equations	p. 355
7.3.3 Finite Element Models	p. 356
7.4 Solid Mechanics	p. 357
7.4.1 Introduction	p. 357
7.4.2 Continuum Equations	p. 357
7.4.3 Constitutive Relations	p. 360
7.4.4 Boundary Conditions	p. 361
7.4.5 Finite Element Models	p. 361
7.4.6 Solution Methods - Quasi-Static Solid Mechanics	p. 363
7.5 Electromagnetics	p. 363
7.5.1 Introduction	p. 363
7.5.2 Maxwell's Equations	p. 364
7.5.3 Electromagnetic Potentials	p. 367
7.5.4 Boundary and Interface Conditions	p. 370
7.5.5 Gauge Conditions	p. 372
7.5.6 Static Field Problems	p. 372
7.5.7 Finite Element Models for EM Fields	p. 374
7.5.8 Solution Methods - EM Fields	p. 379
7.6 Coupled Problems in Mechanics	p. 380
7.6.1 Introduction	p. 380
7.6.2 Heat Conduction - Viscous Fluid Interactions 1 and 2	p. 381
7.6.3 Heat Conduction - Quasi-Static Interactions 1 and 3	p. 381
7.6.4 Heat Conduction - Electric Field Interactions 1 and 4	p. 383
7.6.5 Heat Conduction - Electromagnetic Field Interactions 1 and 4 and 5	p. 384
7.6.6 Viscous Flow - Quasi-Static Solid Interactions 2 and 3	p. 386
7.6.7 Viscous Flow - Electric Field Interactions 2 and 4	p. 387
7.6.8 Viscous Flow - Electromagnetic Field Interactions 2 and 4 and 5	p. 388
7.6.9 Quasi-Static Solid - Electromagnetic Field Interactions 3 and 4 and 5	p. 389
7.7 Implementation of Coupled Algorithms	p. 390
7.8 Numerical Examples	p. 392
7.8.1 Introduction	p. 392
7.8.2 Thermal-Stress Example	p. 392
7.8.3 Thermal-Electromagnetic Example	p. 395
7.8.4 Fluid-Solid Interaction Example	p. 398
7.8.5 Fluid-Electromagnetic Example	p. 400
References for Additional Reading	p. 402
8. Advanced Topics: Parallel Processing	p. 405
8.1 Introduction	p. 405
8.2 Parallel Systems	p. 406
8.2.1 Classification	p. 406
8.2.2 Languages and Communication Utilities	p. 408
8.2.3 Performance	p. 409
8.3 FEM and Parallel Processing	p. 411
8.3.1 Preliminary Comments	p. 411
8.3.2 Generic FEM Steps	p. 411
8.3.3 External Preprocessing	p. 412
8.3.4 Internal Preprocessing	p. 414
8.3.5 Solution Processing	p. 414
8.3.6 Internal Postprocessing	p. 418
8.3.7 External Postprocessing	p. 419
8.3.8 Other Parallel Issues	p. 419
8.4 Summary	p. 421
References for Additional Reading	p. 421
Appendix A Computer Program FEM2DHT	p. 425
A.1 Introduction	p. 425
A.2 Heat Transfer and Related Problems	p. 425
A.3 Flows of Viscous Incompressible Fluids	p. 426
A.4 Description of the Input Data	p. 426
A.5 Source Listings of Selective Subroutines	p. 436
Reference	p. 436
Appendix B Solution of Linear Equations	p. 443
B.1 Introduction	p. 443
B.2 Direct Methods	p. 444
B.3 Iterative Methods	p. 445
B.3.1 General Comments	p. 445
B.3.2 Solution Algorithms	p. 446
References for Additional Reading	p. 450
Appendix C Fixed Point Methods and Contraction Mappings	p. 455
C.1 Fixed Point Theorem	p. 455
C.2 Chord Method	p. 457
C.3 Newton's Method	p. 457
C.4 The Newton-Raphson Method	p. 458
C.5 Descent Methods	p. 459
References for Additional Reading	p. 459
Subject Index	p. 461

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